In this paper, some of the relationships between optimal control and trajectory planning are examined. When planning trajectories for linear control systems, a demand that arises naturally in air traffic control or noise contaminated data interpolation is that the curve passes close to given points, or through intervals, at given times. In this paper, we produce these curves by solving an optimal control problem for linear control systems, while driving the output of the system close to the waypoints. We furthermore show how this optimal control problem reduces to a finite, quadratic programming problem, and we thus provide a constructive. yet theoretically sound framework for producing a rich set of curves called smoothing splines.